The Bridges of Königsberg
Königsberg was a city in Prussia situated on the Pregel River, which served as the residence of the dukes of Prussia in the 16th century. (Today, the city is named Kaliningrad, and is a major industrial and commercial center of western Russia.) The river Pregel flowed through the town, creating an island, as in the following picture. Seven bridges spanned the various branches of the river, as shown.
A famous problem concerning Königsberg was whether it was possible to take a walk through the town in such a way as to cross over every bridge once, and only once. An example of a failed attempt to take such a walking tour is shown below
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To solve this problem :
Suppose they had decided to build one fewer bridge in Konigsberg, so that the map looked like this:
This one is solvable. Here's one possible solution:
In other way soluton using euler solution :
Graphs
Euler's approach was to regard the spots of land (there are 4 of them) as points to be visited, and the bridges as paths between those points. The mathematical essentials of the map of Königsberg can then be reduced to the following diagram.
A graph is a figure consisting of points (called vertices--the plural of vertex) and connecting lines or curves (called edges). The problem of the bridges of Königsberg can then be reformulated as whether this graph can be traced without tracing any edge more than once.
For each of the vertices of a graph, the order of the vertex is the number of edges at that vertex. Euler's Solution
Euler's solution to the problem of the Königsberg bridges involved the observation that when a vertex is "visited" in the middle of the process of tracing a graph, there must be an edge coming into the vertex, and another edge leaving it; and so the order of the vertex must be an even number. This must be true for all but at most two of the vertices--the one you start at, and the one you end at, and so a connected graph is traversible if and only if it has at most two vertices of odd order. (Note that the starting and ending vertices may be the same, in which case the order of every vertex must be even.) Now a quick look at the graph above shows that there are more than two vertices of odd order, and so the graph cannot be traced; that is the desired walking tour of Königsberg is impossible.
Norsulianaeliani Abdul Wahab
Nur Aimi Abdullah Sani 21885
Königsberg was a city in Prussia situated on the Pregel River, which served as the residence of the dukes of Prussia in the 16th century. (Today, the city is named Kaliningrad, and is a major industrial and commercial center of western Russia.) The river Pregel flowed through the town, creating an island, as in the following picture. Seven bridges spanned the various branches of the river, as shown.
A famous problem concerning Königsberg was whether it was possible to take a walk through the town in such a way as to cross over every bridge once, and only once. An example of a failed attempt to take such a walking tour is shown below
1.
2.
3.
To solve this problem :
Suppose they had decided to build one fewer bridge in Konigsberg, so that the map looked like this:This one is solvable. Here's one possible solution:
In other way soluton using euler solution :
Graphs
Euler's approach was to regard the spots of land (there are 4 of them) as points to be visited, and the bridges as paths between those points. The mathematical essentials of the map of Königsberg can then be reduced to the following diagram.
A graph is a figure consisting of points (called vertices--the plural of vertex) and connecting lines or curves (called edges). The problem of the bridges of Königsberg can then be reformulated as whether this graph can be traced without tracing any edge more than once.
For each of the vertices of a graph, the order of the vertex is the number of edges at that vertex.
Euler's Solution
Euler's solution to the problem of the Königsberg bridges involved the observation that when a vertex is "visited" in the middle of the process of tracing a graph, there must be an edge coming into the vertex, and another edge leaving it; and so the order of the vertex must be an even number. This must be true for all but at most two of the vertices--the one you start at, and the one you end at, and so a connected graph is traversible if and only if it has at most two vertices of odd order. (Note that the starting and ending vertices may be the same, in which case the order of every vertex must be even.) Now a quick look at the graph above shows that there are more than two vertices of odd order, and so the graph cannot be traced; that is the desired walking tour of Königsberg is impossible.
Norsulianaeliani Abdul Wahab
Nur Aimi Abdullah Sani 21885